An Online Feasible Point Method for Benign Generalized Nash Equilibrium Problems

TL;DR

The paper tackles the problem of repeated generalized Nash games with endogenous, time-varying shared constraints, where feasibility at every iteration is crucial. It introduces the Online Feasible Point Method (online FPM) with alternating coordination, enabling simultaneous updates while guaranteeing joint feasibility under a class of benign GNEP. The authors establish feasibility guarantees and, for strongly benign GNEP, convergence to the unique GNE; they also connect the framework to online regret minimization, deriving sublinear regret and constraint-violation bounds in the benign setting. The work advances practical coordination in multi-agent systems such as electricity markets and pollution control, where maintaining feasibility and convergence under limited communication is essential. It also outlines broad future directions, including simultaneous coordination, extensions beyond benign GNEP, and richer cooperation/competition dynamics.

Abstract

We consider a repeatedly played generalized Nash equilibrium game. This induces a multi-agent online learning problem with joint constraints. An important challenge in this setting is that the feasible set for each agent depends on the simultaneous moves of the other agents and, therefore, varies over time. As a consequence, the agents face time-varying constraints, which are not adversarial but rather endogenous to the system. Prior work in this setting focused on convergence to a feasible solution in the limit via integrating the constraints in the objective as a penalty function. However, no existing work can guarantee that the constraints are satisfied for all iterations while simultaneously guaranteeing convergence to a generalized Nash equilibrium. This is a problem of fundamental theoretical interest and practical relevance. In this work, we introduce a new online feasible point method. Under the assumption that limited communication between the agents is allowed, this method guarantees feasibility. We identify the class of benign generalized Nash equilibrium problems, for which the convergence of our method to the equilibrium is guaranteed. We set this class of benign generalized Nash equilibrium games in context with existing definitions and illustrate our method with examples.

Figures (3)

• Figure 1: The first 24 iterations of the online FPM Algorithm \ref{['alg:explore']} on the GNEP defined in Example \ref{['Example:SB']}.
• Figure 2: The first 40 iterations of the online FPM Algorithm \ref{['alg:explore']} on a GNEP that does not satisfy Condition 1 in Definition \ref{['def:StronglyBenignGNEP']}. Details in Example \ref{['Example:nB1']}.
• Figure 3: The first 30 iterations of the online FPM on the GNEP defined in Example \ref{['Example:nB2']} for two different initializations.

Theorems & Definitions (34)

• Definition 2.1: Approximate Generalized Nash Equilibrium (GNE)
• Definition 2.2: Strongly Benign GNEP
• Definition 2.3: Benign GNEP
• Definition 2.4: Quasi Variational Inequality
• Proposition 2.1
• Theorem 4.1
• Theorem 4.2
• Example 4.1
• Theorem 5.1
• Example A.1: $(\phi,\delta)$-strongly benign GNEP